Question

The number of values of $\alpha$ in $[0,2\pi ]$ for which $2{\sin }^3\alpha -7{\sin }^2\alpha +7\sin \alpha =2$, is :

• A. $6$
• B. $4$
• C. $3$
• D. $1$

Answer 1

The correct option is C $3$

$2{\sin }^3\alpha -7{\sin }^2\alpha +7\sin \alpha =2$
$2{\sin }^3\alpha -7{\sin }^2\alpha +7\sin \alpha -2=0$
$2(\sin \alpha -1)({\sin }^2\alpha +\sin \alpha +1)-7\sin \alpha (\sin \alpha -1)=0$
$\sin \alpha =1$ or $\sin \alpha =\frac{1}{2}\sin \alpha =2$
Value '$2$' is not possible
So, total $3$ solutions $\alpha =\frac{\pi }{6},\frac{5\pi }{6},\frac{\pi }{2}$.